The origin of ultrahigh piezoelectricity in relaxor-ferroelectric solid solution crystals

The discovery of ultrahigh piezoelectricity in relaxor-ferroelectric solid solution single crystals is a breakthrough in ferroelectric materials. A key signature of relaxor-ferroelectric solid solutions is the existence of polar nanoregions, a nanoscale inhomogeneity, that coexist with normal ferroelectric domains. Despite two decades of extensive studies, the contribution of polar nanoregions to the underlying piezoelectric properties of relaxor ferroelectrics has yet to be established. Here we quantitatively characterize the contribution of polar nanoregions to the dielectric/piezoelectric responses of relaxor-ferroelectric crystals using a combination of cryogenic experiments and phase-field simulations. The contribution of polar nanoregions to the room-temperature dielectric and piezoelectric properties is in the range of 50–80%. A mesoscale mechanism is proposed to reveal the origin of the high piezoelectricity in relaxor ferroelectrics, where the polar nanoregions aligned in a ferroelectric matrix can facilitate polarization rotation. This mechanism emphasizes the critical role of local structure on the macroscopic properties of ferroelectric materials.


Supplementary Note 1: Analysis of frequency-dependent transverse dielectric responses for relaxor-PT crystals.
The frequency-dependent dielectric permittivity and loss of PMN-0.32PT and PZN-0.15PT crystals are shown in Supplementary Figure 8. In the frequency range of 10 -1 -10 5 Hz, frequency dependent behaviors were clearly observed at the temperature below 200 K.
According to Debye theory, the relaxational dielectric spectrum can be described by means of the distribution function of the relaxation time g(τ) 6 : (1) where g(τ) is normalized by the following equation: (2) In this study, we assume there is a strong Debye relaxation with a constant distribution between certain upper value (τ1) and lower value (τ2). Thus, the g(τ) is assumed to be: In this condition, ∞ represents the dielectric permittivity at frequency much higher than 1/τ2. Δ represents the contribution of the relaxation to the static permittivity. Based on Eqs. 1-3, the frequency dependent dielectric permittivity can be expressed as: The dielectric spectra of PMN-0.32PT and PZN-0.15PT crystals are fitted by using Eq. 4 and given in Supplementary Figure 9. It can be seen that the upper frequency limit is around 10 7 -10 8 for PZN-0.15PT and PMN-0.32PT, being much lower than the frequency of phonon modes (10 12 -10 14 Hz). The observed relaxation frequency range (10 0 -10 8 ) is similar to that of dipole reorientation. This indicates that some interface motion or dipoles' switching significantly contributes to the dielectric response at 120 K and 150 K for relaxor-PT crystals.

Supplementary Note 2: Inference of the size of PNRs by Arrhenius law.
Supplementary Figure 10 a, b and c show the temperature dependence of the imaginary part (ε″/ε0) of the transverse dielectric permittivity for PMN-0.28PT, PMN-0.32PT and PZN-0.15PT crystals, respectively. Supplementary Figure 10 d, e and f show the temperature of maximum ε″/ε0 versus frequency. It can be seen that ln(f) and 1/Tm follow a linear relationship, which can be expressed by Arrhenius law: where f is the measurement frequency, Ea the activation energy, kB the Boltzmann constant, f0 the attempt frequency, and Tm the temperature of maximum ε″/ε0. By fitting Eq. 5, it is found that the activation energy Ea is around 2000kB for PMN-0.28PT, PMN-0.32PT and PZN-0.15PT crystals. Activation energy Ea can be expressed as ΔGV, where ΔG is the energy barrier density for reorientation of a PNR and V the volume of a single PNR. According to Landau-Devonshire formalism of free energy 2,7 , the typical values of ΔG are in the range of 10 5~1 0 7 J m -3 . Based on this typical value of ΔG, the volume of a PNR (V) is calculated to be in the range of (27.4~274)×10 -27 m 3 .
Thus, the size of PNRs ( √ 3 ) is inferred to be in the range of 1.4~6.5 nm.

(1) Framework of the phase field model
The temporal evolution of a polarization field is described by the Time-Dependent Ginzburg-Landau (TDGL) equation, where L is the kinetic coefficient, F, the total free energy of the system, and ) , ( t P i r is the polarization. ) , is the thermal dynamic driving force for the spatial and temporal evolution of ) , ( t P i r  . ) , ( t i r  represents the effect of thermal noise, which is a Gaussian random fluctuation satisfying: where T is the temperature, m represents the amplitude of thermal noise, δij is the Kronecker delta, ) ' ( r r   and are the delta function. The bracket <…> denotes an average value. The total free energy of the system includes the bulk free energy, elastic energy, electrostatic energy, and the gradient energy: (8) where V is the system volume of the PNR-ferroelectric composite, fbulk denotes the Landau bulk free energy density, felas the elastic energy density, felec the electrostatic energy density and fgrad the gradient energy density.
The bulk free energy density is expressed by Landau theory, i.e.
where α1, α11, α12, α111, α112 and α123 are Landau energy coefficients. The values of these coefficients determine the thermodynamic behaviors of the bulk phases (paraelectric or ferroelectric). In our simulation work, the difference between the ferroelectric matrix and PNRs are reflected in the fbulk, where different Landau coefficients were used for matrix and PNRs, respectively.
The gradient energy density, associated with the formation and evolution of domain walls, can be expressed as: where Gij are gradient energy coefficients. Pi,j denote ∂Pi/∂rj.
The corresponding elastic energy density can be expressed as: where cijkl is the elastic stiffness tensor, εij the total strain, the electrostrictive stress-free strain, i.e., .
The electrostatic energy density is given by: where is the E-field induced by the dipole moments in the specimen. The detailed expression of can be found in Ref. 8. is an applied external E-field.

(2) Material constants and numerical simulation parameters of PNR-ferroelectric composite
A semi-implicit Fourier-spectral method is adopted for numerically solving the TDGL equation 9 . Based on the experimental data (polarization and dielectric response) and the parameters of typical perovskite ferroelectrics (Pb(Zr,Ti)O3 and BaTiO3) 2,10 , the following parameters are adopted in the present simulation. For the ferroelectric matrix, the Landau free energy parameters are: , , , , , . In the whole temperature range, there is only a Cubic-Tetragonal phase transition at 468 K for the ferroelectric matrix. The spontaneous polarization of matrix is about 0.39 C m -2 at 300 K, being similar to the experimental value of the PZN-0.15PT crystal. At 20 K, the dielectric permittivity ε11/ε0 and ε33/ε0 are 850 and 160 respectively for ferroelectric matrix, being similar to the measured data of PZN-0.15PT (At temperature of 20 K and frequency of 100 Hz, the measured ε11/ε0 and ε33/ε0 are 930 and 250, respectively.) For PNRs, the Landau free energy parameters are:  ,  ,  ,  ,  , , where T is the temperature in Kelvin. In the whole temperature range, there is only a Cubic-orthorhombic phase transition around 620 K for PNRs. The spontaneous polarization of PNR is 0.41 C m -2 at 300 K without consideration of interactions from the matrix. The elastic constants and electrostrictive coefficients are set to be the same for the matrix (the elastic constants refer to the data of PZN-0.12PT crystal 11 and the electrostrictive coefficients refer to data for PbTiO3 2 ). The background dielectric permittivity (kb), associated with the contribution of the hard mode to the permittivity 12 , is set to be 100×ε0. In the computer simulations, we employed 2D 128×128 discrete grid points (3D: 64×64×64 grid points) and periodic boundary conditions. The grid space in real space is chosen to be . The gradient energy coefficients are chosen to be , , , where 13 . Based on these parameters, the simulated width of domain walls was found to be 1-2 nm, being consistent with existing experimental measurements 14,15 . The time step for integration is , where . For 2D simulation, Pz is set at zero.

Supplementary Note 4: The effect of PNRs volume fraction on the transverse dielectric response for the PNR-ferroelectric composite.
The temperature dependent dielectric permittivities were simulated under different PNR volume fractions for a PNR-ferroelectric composite, as shown in Supplementary Figure 12. It was found that the increase of volume fraction of PNRs may bring two effects. First, higher volume fraction of PNRs may result in higher dielectric response when most of the PNRs are unstable or in "collinear" state. Secondly, higher volume fraction of PNRs could result in stronger interaction among PNRs due to the decrease of average distances among PNRs. Thus, PNRs are more stable for higher volume fractions, where higher temperatures are needed to make PNRs unstable and/or in "collinear" state. As shown in Supplementary Figure 12, a composite with higher volume fraction of PNRs shows higher dielectric response at elevated temperature region, but at the cost of temperature stability.

Supplementary Note 5: The effect of thermal fluctuations on the dielectric responses for the PNR-ferroelectric composite.
PNR-ferroelectric composite at different levels of thermal fluctuation. Over all, the temperature dependent properties aren't affected significantly by thermal noise, though a bit higher fluctuation is observed for the magnitude of 5.